Classical Nucleation Theory - Aerosols: Measurement, Dosimetry, and Health Effects - page 509

19.4 TRANSLATION-ROTATION CORRECTION FACTOR
Frenkel [1] was the irst who argued that the contribution to the free energy of the critical nucleus
from the translational and rotational degrees of freedom should be taken into account when cal-
culating the drop (cluster) size distribution. To determine the equilibrium cluster size distribution,
Frenkel has considered an ensemble of clusters as an ideal gas mixture. In this case a statistical
mechanical analysis gives [1,11,32]

e v μ /

n k T

N = n q (19.25)

B

n

where
N n is the equilibrium number of n -sized clusters (consisting of n monomeric molecules)
q n is the partition function within the canonical ensemble for the drop containing n molecules
μ v is the chemical potential for the vapor molecules

According to Frenkel's model [1],

Q Q Q Q

tr rot tr rot

q = Q e

− U k T /

= q

rest

(19.26)

n B

n 3 , n v n

6 6

( Q ) ( Q )

v v

where
Q tr and Q rot are the partition functions for three translational and three rotational degrees of
freedom of n -sized cluster, respectively
Q 3n,v is the partition function for 3 n vibrational degrees of freedom
( Q v ) 6 is the partition function for six vibrational degrees of freedom which are to be deactivated

All the clusters in Frenkel's model have the same structure corresponding to the minimum potential
energy U n . q rest in Equation 19.26 is the partition function for the cluster at rest [38]:

n rest

q

rest

= Q e

− U k T /

= e

− f / k T

(19.27)

n B B

n 3 , n v

where f n rest is the Helmholtz free energy for the cluster at rest. According to the simple Frenkel's
model the free energy f n rest is assumed to be given by [1]

f

rest
n

= 0 f n + γ n

2 3 /

(19.28)

where
f 0 is the Helmholtz free energy per one molecule in the bulk liquid phase
γ is a constant proportional to the surface tension of the lat interface

We refer to the factor Q tr Q rot /( Q v ) 6 in Equation 19.26 as the Frenkel factor. Equation 19.28 is the basis
of the CNT. In the framework of CNT the embryo of the nucleating phase is regarded as a spherical
incompressible liquid drop ixed in the space; the density of the drop is considered as homogeneous
and equal to that of the bulk liquid; this drop has a sharply deined interface with the surrounding
metastable mother phase which, in the case of vapor, is regarded as an ideal gas; the surface tension
of the critical nucleus is regarded as equal to that of the lat interface (capillarity approximation).
Using instead of Equation 19.28 a similar formula for the Gibbs free energy g n = μ l n + γ n 2/3 (where
μ l is the chemical potential of a molecule, as if it were part of a bulk liquid at the pressure P outside
the drop) Frenkel has derived a rigorous thermodynamic formula for the cluster (drop) equilibrium
size distribution. This distribution proves to be [1]